代数整数环Z[ω]的素元及剩余类环

Simple Element and Residue Class Ring of Algebraic Integer Ring Z[ω]

作为抽象代数中环理论的两个重要环Z[i]与Z[ω],常以特例的形式散见于抽象代数教材中,对其系统的讨论不多见.而这两个环不仅是抽象代数中的重要实例,而且它们的性质是数论中相关理论的重要基础,特别是在解决费马问题n=3的情形时发挥了关键的作用.文中较为系统的讨论了整环Z[ω],确定了Z[ω]中的素元及其剩余类环所含元素的个数,由此得到数论中一个与Fermat小定理类似的结果.

Z[i] and Z[ω], two important rings in the ring theory of abstract algebra, rarely make their appearance in textbooks, their systematic discussion is not much talked about. Yet, they are of great significance, besides, their properties are the important basis for the relevant theory in the number theory, in particular, Z[ω] plays a very important role in solving Fermat＇s conjecture n = 3. This paper gives a systematic discussion of integer ring Z[ω], and determines the elements included in the residue class ring. In the number theory, the results are similar to those of Fermat＇s theorem.